The Julia sets of Chebyshev's method with small degrees

2022/01/26 by Tarakanta Nayak, Nayak, Tarakanta, Soumen Pal +1 · 2 citations
Mathematics · #37F10 #65H05 #Dynamical Systems (math.DS) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2201.11055

openalex publication_date 2022/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a polynomial p, the degree of its Chebyshev's method C_p is determined. If p is cubic then the degree of C_p is found to be 4,6 or 7 and we investigate the dynamics of C_p in these cases. If a cubic polynomial p is unicritical or non-generic then, it is proved that the Julia set of C_p is connected. The family of all rational maps arising as the Chebyshev's method applied to a cubic polynomial which is non-unicritical and generic is parametrized by the multiplier of one of its extraneous fixed points. Denoting a member of this family with an extraneous fixed point with multiplier λ by C_λ, we have shown that the Julia set of C_λ is connected whenever λ∈ [-1,1].

Cited by

• Chebyshev's method applied to polynomials with two distinct roots
• On the dynamics of Halley's method

Related

• Symmetry and dynamics of Chebyshev's method
• Chebyshev's method applied to polynomials with two distinct roots
• Chebyshev's method for exponential maps
• Connectivity of the Julia set for the Chebyshev-Halley family on degree n polynomials
• Newton-like components in the Chebyshev-Halley family of degree n polynomials